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Chapter 1: Problem 17

Simplify the quantities using \(m(z)=z^{2}\). $$m(z)-m(z-h)$$

Short Answer

Expert verified
The simplified expression is \(2zh - h^2\).

Step by step solution

01

Understand the Function

We start by understanding that the function given is \(m(z) = z^2\). This is a basic quadratic function where \(z\) is squared.
02

Evaluate \(m(z-h)\)

Substitute \(z-h\) into the function \(m(z)\):\[ m(z-h) = (z-h)^2 \]Expand \((z-h)^2\) using the binomial theorem:\[ m(z-h) = z^2 - 2zh + h^2 \]
03

Calculate \(m(z) - m(z-h)\)

Subtract \(m(z-h)\) from \(m(z)\):\[ m(z) - m(z-h) = z^2 - (z^2 - 2zh + h^2) \]Simplify the expression by distributing the minus sign:\[ m(z) - m(z-h) = z^2 - z^2 + 2zh - h^2 \]Cancel out \(z^2\) terms:\[ m(z) - m(z-h) = 2zh - h^2 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quadratic Functions
A quadratic function is a specific type of polynomial function that can be represented by the form \( y = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \). In the exercise, we focused on the function \( m(z) = z^2 \). Here, the coefficient \( a \) is 1, and there are no linear \( b \) or constant \( c \) terms. The key characteristic of any quadratic function is its parabolic graph, which can open upwards or downwards depending on the sign of \( a \).
  • The vertex of the parabola represents either a maximum or minimum point.
  • For the specific quadratic \( z^2 \), the graph opens upwards with a vertex at the origin (0,0).
Understanding how quadratic functions behave is essential in various mathematical problems, including simplifying differences like those in the exercise.
Binomial Theorem
The binomial theorem provides a way to expand expressions of the form \((x + y)^n\). In the given problem, we used this theorem to expand \((z-h)^2\). According to the binomial theorem, this expansion is given by:
  • \( (z-h)^2 = z^2 - 2zh + h^2 \)
This formula comes from applying the theorem, which states that any power of a binomial can be expanded as:
\[ (x + y)^2 = x^2 + 2xy + y^2 \] In our exercise, due to the subtraction of \( h \), the terms follow a similar pattern but with negative signs. The binomial theorem makes it easy to expand squared terms and is a valuable tool for polynomial simplification. It streamlines complex algebraic expressions into simpler forms.
Algebraic Simplification
Algebraic simplification involves reducing complex expressions into simpler or more manageable forms. This is often achieved by combining like terms, factoring, or performing operations such as distribution. In the exercise, we started with simplifying \( m(z) - m(z-h) \).
  • Initial expression was \( z^2 - (z^2 - 2zh + h^2) \).
  • Applying the distributive law by removing parentheses and distributing the negative sign results in \( z^2 - z^2 + 2zh - h^2 \).
  • The terms \( z^2 - z^2 \) cancel each other out.
This simplification process left us with \( 2zh - h^2 \), a much simpler expression that is easy to work with. Practicing algebraic simplification helps improve problem-solving skills and transforms complex problems into more understandable ones.

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Most popular questions from this chapter

Is the statement true or false? Assume that \(\lim _{x \rightarrow 7^{+}} f(x)=2\) and \(\lim _{x \rightarrow 7^{-}} f(x)=-2 .\) Explain. The function \((f(x))^{2}\) is continuous at \(x=7\).

The voltage, \(V,\) of an electrical outlet in a home as a function of time, \(t\) (in seconds), is \(V=V_{0} \cos (120 \pi t)\) (a) What is the period of the oscillation? (b) What does \(V_{0}\) represent? (c) Sketch the graph of \(V\) against \(t\). Label the axes.

Give an explanation for your answer. The graph of \(g(x)=\log (x-1)\) crosses the \(x\) -axis at \(x=1.\)

Explain what is wrong with the statement. $$\ln (A+B)=(\ln A)(\ln B).$$

Different isotopes (versions) of the same element can have very different half-lives. With \(t\) in years, the decay of plutonium- 240 is described by the formula $$Q=Q_{0} e^{-0.00011 t},$$ whereas the decay of plutonium-242 is described by $$Q=Q_{0} e^{-0.0000018 t}.$$ Find the half-lives of plutonium-240 and plutonium-242.
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